Given three vectors `veca, vecb` and `vecc` are non-zero and non-coplanar vectors. Then which of the following are coplanar.

Options

A. `veca+vecb, vecb+vecc,vecc+veca`
B. `veca-vecb,vecb+vecc,vecc+veca`
C. `veca+vecb,vecb-vecc,vecc-veca`
D. `veca+vecb,vecb+vecc,vecc-veca`

Explanation

To determine which combinations of the vectors \( \vec{a}, \vec{b}, \) and \( \vec{c} \) are coplanar, we will analyze the given options step by step.

### Step 1: Understanding Coplanarity
Three vectors are coplanar if they lie in the same plane. This can be determined if one vector can be expressed as a linear combination of the others.

### Step 2: Analyze Each Option
We will evaluate the combinations provided in the question.

#### Option 1: \( \vec{a} + \vec{b}, \vec{b} + \vec{c}, \vec{c} + \vec{a} \)
- We need to check if these vectors can be expressed in terms of each other.
- There is no direct linear combination that can express one of these vectors in terms of the others.
- Therefore, these vectors are **not coplanar**.

#### Option 2: \( \vec{c} + \vec{a}, \vec{b} + \vec{c}, \vec{a} + \vec{b} \)
- We can express \( \vec{c} + \vec{a} \) as \( (\vec{b} + \vec{c}) + (\vec{a} - \vec{b}) \).
- This shows that \( \vec{c} + \vec{a} \) can be represented as a combination of the other two vectors.
- Hence, these vectors are **coplanar**.

#### Option 3: \( \vec{a} + \vec{b}, \vec{c} - \vec{a}, \vec{b} - \vec{c} \)
- We can express \( \vec{a} + \vec{b} \) as \( (\vec{c} - \vec{a}) + (\vec{b} - \vec{c}) + \vec{c} \).
- This shows that \( \vec{a} + \vec{b} \) can be represented as a combination of the other two vectors.
- Hence, these vectors are **coplanar**.

#### Option 4: \( \vec{c} - \vec{a}, \vec{b} + \vec{c}, \vec{a} + \vec{b} \)
- We can express \( \vec{c} - \vec{a} \) as \( (\vec{b} + \vec{c}) - (\vec{a} + \vec{b}) + \vec{b} \).
- This shows that \( \vec{c} - \vec{a} \) can be represented as a combination of the other two vectors.
- Hence, these vectors are **coplanar**.

### Conclusion
From our analysis, the coplanar combinations are:
- Option 2: \( \vec{c} + \vec{a}, \vec{b} + \vec{c}, \vec{a} + \vec{b} \)
- Option 3: \( \vec{a} + \vec{b}, \vec{c} - \vec{a}, \vec{b} - \vec{c} \)
- Option 4: \( \vec{c} - \vec{a}, \vec{b} + \vec{c}, \vec{a} + \vec{b} \)

### Final Answer
The coplanar vectors are from options 2, 3, and 4.

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